reserve x, y, z, v for set,
  n, m, k for Nat;
reserve G, G1, G2, G3 for Graph;
reserve x, y for Element of (the carrier of G);

theorem
  for G1,G2 being strict Graph holds
  (the Source of G1) tolerates (the Source of G2) &
  (the Target of G1) tolerates (the Target of G2) &
  bool (G1 \/ G2) c= (bool G1) \/ (bool G2) implies G1 c= G2 or G2 c= G1
proof
  let G1,G2 be strict Graph;
  assume
  A1: (the Source of G1) tolerates (the Source of G2) & (the Target of G1)
  tolerates (the Target of G2);
  assume
A2: bool (G1 \/ G2) c= (bool G1) \/ (bool G2);
A3: G1 \/ G2 in bool (G1 \/ G2) by Th29;
A4: now
    assume G1 \/ G2 in bool G1;
then  G1 \/ G2 is Subgraph of G1 by Def25;
then A5: G1 \/ G2 c= G1;
 G1 c= G1 \/ G2 by A1,Th19;
then  G1 \/ G2 = G1 by A5,Th16;
    hence G2 c= G1 by A1,Th24;
  end;
 now
    assume G1 \/ G2 in bool G2;
then  G1 \/ G2 is Subgraph of G2 by Def25;
then A6: G1 \/ G2 c= G2;
 G2 c= G1 \/ G2 by A1,Th19;
then  G1 \/ G2 = G2 by A6,Th16;
    hence G1 c= G2 by A1,Th24;
  end;
  hence thesis by A2,A3,A4,XBOOLE_0:def 3;
end;
